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Thread: Convergent sequence

  1. #1
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    Convergent sequence

    I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

    Suppose sequence $\displaystyle a_n > 0$ and $\displaystyle b_n = a_n + \frac{1}{a_n}
    $
    Assume $\displaystyle a_n$ >= 1 for all n, and that $\displaystyle b_n$ converges. Prove that $\displaystyle a_n $ converges.

    If it assumed that $\displaystyle b_n$ converges but only$\displaystyle a_n > 0$, it does not follow that $\displaystyle a_n$ converges. Find the example.
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  2. #2
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    Quote Originally Posted by Rozaline View Post
    I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

    Suppose sequence $\displaystyle a_n > 0$ and $\displaystyle b_n = a_n + \frac{1}{a_n}
    $
    Assume $\displaystyle a_n$ >= 1 for all n, and that $\displaystyle b_n$ converges. Prove that $\displaystyle a_n $ converges.

    If it assumed that $\displaystyle b_n$ converges but only$\displaystyle a_n > 0$, it does not follow that $\displaystyle a_n$ converges. Find the example.
    If $\displaystyle b_n = a_n + \tfrac{1}{a_n}$ then $\displaystyle a_n^2 - b_na_n + 1 = 0$. Use the quadratic formula to solve that equation: $\displaystyle a_n = \tfrac12\bigl(b_n\pm\sqrt{b_n^2-4}\bigr)$. That should tell you enough about $\displaystyle a_n$ to find solutions for both parts of the problem.
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